Viterbo's conjecture for Lagrangian products in $\mathbb{R}^4$ and symplectomorphisms to the Euclidean ball

Abstract

We use the generalized Minkowski billiard characterization of the EHZ-capacity of Lagrangian products in order to reprove that the $4$-dimensional Viterbo conjecture holds for the Lagrangian products (any triangle/parallelogram in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) and extend this fact to the Lagrangian products (any trapezoid in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$). Based on this analysis, we classify equality cases of this version of Viterbo's conjecture and prove that most of them can be proven to be symplectomorphic to Euclidean balls. As a by-product, we prove sharp systolic Minkowski billiard / worm problem inequalities. Furthermore, we discuss the Lagrangian products (any convex quadrilateral in $\mathbb{R}^2$)$\times$(any convex body in $\mathbb{R}^2$) for which we show that the truth of Viterbo's conjecture would follow from the positive solution of a challenging Euclidean covering problem. Finally, we show that the flow associated to equality cases of Viterbo's conjecture for Lagrangian products in $\mathbb{R}^4$--which turn out to be convex polytopes--is not Zoll in general, but that a weaker Zoll property, namely, that every characteristic almost everywhere away from lower-dimensional faces is closed and action-minimizing, does apply.